Riemannian and Sub-Riemannian Geodesic Flows fileRiemannianandSub-RiemannianGeodesicFlows MauricioGodoyMolina1 Joint with E. Grong Universidad de La Frontera (Temuco, Chile) Potsdam,February2017

Riemannian and Sub-Riemannian Geodesic Flows

Mauricio Godoy Molina1Joint with E. Grong

Universidad de La Frontera (Temuco, Chile)

Potsdam, February 2017

1Partially funded by grant Anillo ACT 1415 PIA CONICYT (Chile)Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 1 / 24

Outline

1 Introduction: Geometry and restrictions

2 Geodesic Flows

3 Results

Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 2 / 24

Philosophy: Restrictions in mechanics

In classical analytic mechanics we study systems of the form

Differential equation + restrictions.

Usually: restrictions on the position and/or velocity, but

restr. on the velocity can “hide” restr. on the position.

Definition (heuristic)Restrictions on the position (hidden or not) are called holonomic.

In what follows, we call non-holonomic to those restrictions on velocity nothiding restrictions on the position.


Hidden restrictions

In examples 1-3: a particle P moves in R3 with position −→r = −→r (t),velocity −→v = −→v (t) and under certain restrictions.

Example 1: −→v ⊥ −→r ⇐⇒P moves on the sphere of radius |−→r (0)| centered at −→0 .

In other words, −→v ⊥ −→r is holonomicsince it “hides a restriction on theposition.”

DefinitionThe restriction on −→v can beintegrated to a restriction on −→r .


Perturbing restrictionsExample 2: −→v ⊥

−→k ⇐⇒ −→v = a · −→ı + b · −→ ⇐⇒

P moves on a horizontal plane.

Holonomic again. But. . . what if wemodify the problem slightly?

Example 3: −→v ⊥(−→k − x−→

)⇐⇒ −→v = a · −→ı + b ·

(−→ + x−→k).

Non-holonomic! Moreover:

Theorem (Folklore)Small perturbations of restrictionsare non-holonomic.


Mathematical formulationPrevious fact: “Most” restrictions are non-holonomic.That’s why we want to understand them.

Space of configurations: Q, with dimQ = dof.Restrictions: −→v (t) ∈ spanX1, . . . ,Xk.

Example 1: Q = sphere, −→v (t) ∈ span∂θ, ∂ϕ

in spherical coordinates.

Example 2: Q = horizontal plane, −→v (t) ∈ span−→ı ,−→ .

Example 3: Q = R3, −→v (t) ∈ span−→ı ,−→ + x

−→k.

DefinitionThe flow of X ∈ Γ(Q) is the function Q × R→ Q given by

eτXp = solution ofγ(s) = X (γ(s))γ(0) = p

in time s = τ.


Detecting holonomy I

Flow of −→ı : eτ−→ı (x , y , z) = (x + τ, y , z).

Flow of −→ : eτ−→ (x , y , z) = (x , y + τ, z).

Flow of −→ + x−→k : eτ(−→ +x

−→k )(x , y , z) = (x , y + τ, z + τx).

Theorem (Frobenius, 1877)A restriction H = spanX1, . . . ,Xk is holonomic iff

ddτ

∣∣∣∣τ=0

e−√τY e−

√τXe

√τY e

√τXp ∈ H, ∀X ,Y ∈ H,∀p ∈ Q.

Explanation?


Detecting holonomy II

For simplicity: t =√τ . Suppose X and Y restrictions in H.

We follow X ,afterward Y ,we come back using X ,finally we come back using Y .

Is the velocity of this new curve a restriction?If yes, then H is holonomic.

DefinitionThe process described above is called commuting flows.


Detecting holonomy III

Example 2: Holonomic

ddτ

∣∣∣∣τ=0

e−√τ−→ e−

√τ−→ı e√τ−→ e√τ−→ı (x , y , z) =

ddτ

∣∣∣∣τ=0

(x , y , z)︸︷︷︸constant!

= (0, 0, 0) ∈ span−→ı ,−→ .

Example 3: Non-holonomic

ddτ

∣∣∣∣τ=0

e−√τ(−→ +x

−→k )e−

√τ−→ı e√τ(−→ +x

−→k )e√τ−→ı (x , y , z) =

ddτ

∣∣∣∣τ=0

(x , y , z + τ) = (0, 0, 1) =−→k /∈ span

−→ı ,−→ + x−→k.


Outline


2 Geodesic Flows

3 Results


Affine control systemsGiven H = spanX1, . . . ,Xk, k ≤ n, we have the control problem

x =k∑

i=1uiXi (x)⇐⇒ x ∈ H. (SAC)

Same question:Given x0, xT ∈ M, can we find u1, . . . , uk such that

x(0) = x0 y x(T ) = xT ?

BIG observation: The controllability of (SAC) is a consequence of Hbeing completely non-holonomic.

DefinitionH is completely non-holonomic if wecan obtain any velocity by commutingflows in H.


Geodesics

Suppose that (SAC) is controllable.

x =k∑

i=1uiXi (x)⇐⇒ x ∈ H (SAC)

ProblemGiven x0, xT ∈ M, we want to find u = (u1, . . . , uk) such that

x(0) = x0, x(T ) = xT and

J(u) = minu

J(u) = minu

12

∫ (u1(t)2 + · · ·+ uk(t)2

)dt.

Usually: A curve γ with control u is called sR-geodesic if k < n andR-geodesic if k = n.


Cometric and metric

Consider a bilinear non-negative tensor h∗ on T ∗M (a cometric) and

]h∗ : T ∗M → T ∗∗M = TM, p 7→ h∗(p, ·)

H = im ]h∗ are the restrictions on M endowed with the metric

h(]h∗p, ]h∗q) = h∗(p, q)

Fact(H,h) determines uniquely h∗. Besides ker ]h∗ = Ann H.


Hamiltonian and flows

The Hamiltonian Hh associated to the metric h is simply

Hh(p) = 12h∗(p, p).

If ω is the canonical 2-form on T ∗M, then

DefinitionThe Hamiltonian vector field ~Hh is given by

dHh(X ) = ω(~Hh,X ), ∀X ∈ X(M).

If h is a (sub-)Riemannian metric, then et~Hh is the (sub-)Riemanniangeodesic flow.


Outline


2 Geodesic Flows

3 Results


Motivating problem I

Theorem (Montgomery)Given a principal G-bundle G y M π→ N (+ technical hypotheses), thenthe normal sub-Riemannian geodesics on M defined by H = (ker dπ)⊥ aregiven by

γsr (t) = expr (tv) · expG(−tA(v)),

where A is the g-valued connection one form.

Idea:Compute Riemannian geodesics in M.Project down to N.“Horizontally lift” the curve to M.


Horizontal lift

Given a submersion π : M → N and a vector v ∈ TxN, then at eachx ∈ π−1(x) there is a unique vector v ∈ TxM such that dxπ(v) = v . Thisis the horizontal lift of v . For a vector field X ∈ Γ(TN), define X by

X |x = X |x .

The horizontal lift γ of γ : [0,T ]→ N at2 x is the unique solution to

˙γ = ¯γ, γ(0) = x

2Obviously πx = γ(0)Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 17 / 24

Motivating problem II

Montgomery’s theorem can be used for some examples.Hopf fibration: U(1) y S2n+1 → CP1

Quaternionic Hopf fibration: Sp(1) y S4n+3 → HPn

Grassmannians: U(n) y Vn,k → Grn,k

BUT there are “true” fibrations: S7 → S15 → OP1 ∼= S8 (octonionicHopf). What to do when there is no group?

Remark (Ornea, Parton, Piccinni, Vuletescu 2013)The situation is worse than expected: ANY v.f. tangent to the leaves ofS7 → S15 → S8 has a zero.


Taming metrics

Let (M,H,h) a sub-Riemannian manifold.A Riemannian metric g on M tames h if g|H = h.

“Natural” questionIs there a relation between the geodesics of g and the ones of h?

Let V = H⊥ with respect to g. Define v = g|V .

Technical toolThe following formula defines a conection (Bott)

∇XY =prH∇gprHX prHY + prV∇

gprVX prVY+

prH[prVX ,prHY ] + prV [prHX , prVY ].


Key Lemma

Lemma (G., Grong)If ·, · denotes the Poisson bracket wrt ω, then

Hh,Hv = Hh,Hg = 0 ⇐⇒ ∇g = 0

If ΠM : T ∗M → M is the canonical projection, then

expsr : Ux ⊆ T ∗x M → M, expsr (x , tp) = (ΠM et~Hh)(p)

expr : Vx ⊆ TxM → M, expr (x , t]p) = (ΠM et~Hg)(p)

Consequence

expr (x , t]p) =(ΠM et~Hh et~Hv)(p) =

(ΠM et~Hv et~Hh)(p).


Main result I

Theorem (G., Grong)(M, g) Riemannian + F tot. geodesic Riem. foliation of Mw/subbundle V. Define (M,H,h) where H = V⊥ and h = g|H. Then, forany x ∈ M and p ∈ T ∗M

expsr (x , tp) = expr (expr (x , t]p),−tprVPt]p) ,

where Pt is the parallel transport along expr (x , t]p)

Compare with Montgomery’s geodesics γsr (t) = expr (tv) · expG(−tA(v)).


Main result II

If we assume that V is integrable, then we know more.

Theorem (G., Grong)Every curve of the form expsr (x , tp) is the horizontal lift of the projectionof the curve expr (x , t]p) iff(a) V is the orthogonal complement of H.(b) The leaves of the foliation of V are totally geodesic.

For the interested few: “Riemannian and Sub-Riemannian GeodesicFlows”. To appear J. Geom. Analysis (I guess this year)


Examples

For principal bundles (+ technical conditions), the formula

expsr (x , tp) = expr (expr (x , t]p),−tprVPt]p)

coincides with Montgomery’s resultThe result can be applied to S7 → S15 → S8, but we have no explicitformulas yet (M.Sc. problem anyone?)If (M,H,h), where H = kerα for α ∈ Ω1(M) contact, then

∇g = 0 iff LZg = 0,

where Z is the Reeb vector field, g(Z ) = 1 and g|H = h



Documents

Riemannian and Sub-Riemannian Geodesic Flows fileRiemannianandSub-RiemannianGeodesicFlows MauricioGodoyMolina1 Joint with E. Grong Universidad de La Frontera (Temuco, Chile) Potsdam,February2017